= Poincaré lemma
{c}
{title2=$d\alpha=0\Longrightarrow\alpha=d\beta\text{ locally},\quad p>0$}
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Every smooth <closed differential form> of positive degree is locally an <exact differential form>. On a coordinate ball centered at zero, put $F_t(x)=tx$ and let $R=x^i\partial_i$ be the radial <vector field>. Define the degree-lowering operator
$$
K\alpha=\int_0^1t^{-1}F_t^*(\iota_R\alpha)\,dt.
$$
For a $p$-form with $p>0$, the integrand has a factor $t^{p-1}$ in its coordinate coefficients and is smooth at zero. <Cartan's magic formula> and differentiation of the <pullback of a differential form> give $\partial_t(F_t^*\alpha)=t^{-1}F_t^*(d\iota_R\alpha+\iota_Rd\alpha)$. Integrating from zero to one gives $dK\alpha+Kd\alpha=\alpha$, since $F_0^*\alpha=0$ in positive degree. If $d\alpha=0$, take $\beta=K\alpha$. The <local potential of a closed differential one-form> gives the explicit degree-one construction. Global exactness can fail through nonzero periods, measured by <de Rham cohomology>.
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